On large scales, our Universe is almost homogeneous and isotropic, but as far as its curvature is concerned we cannot firmly conclude that it is spatially flat [1]}. Even though the observational probes in the past years were in agreement with spatial flatness [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, however, this result has recently been questioned by some of the experiments. The investigations with the Cosmic Microwave Background (CMB) temperature and polarization spectra from Planck 2018 team using the baseline Plik likelihood are suggesting that our Universe could have a closed geometry at more than three standard deviations [8]}, [9]}, [10]}, [11]} (\(4\sigma \)  when the physical curvature density is considered [12]}), and such indications are coming mainly from the temperature data affected by an otherwise inexplicable excess of lensing (\(A_{\rm lens}\)  problem [13]}, [14]}, [11]}). Furthermore, also from the complementary examinations using the alternative CamSpec likelihood [16]}, [17]}, the closed geometry of the Universe is supported at more than \(99 \%\)  CL, and the indication seems to persist even considering the CMB temperature-only data of the new Planck PR4 analysis [18]}. Additionally, an indication for a closed universe is also present in the BAO data, using Effective Field Theories of Large Scale Structure, once the (hidden) assumptions of flatness are removed from the beginning [19]}. These outcomes challenged the assumptions of flatness present in the standard \(\Lambda \) CDM model and sparked a debate about the flatness of our Universe. In fact, if there is no solid theoretical argument, then observations in agreement with a small curvature cannot be used as a proof for a spatially flat universe [20]}, and the assumption of a spatially flat Universe on the spatially curved Universe may significantly affect the cosmological parameters [21]}.
Therefore, in this article we want to investigate, in a systematic way, some well known and classical cosmological models, leaving the curvature of the Universe as a free parameter and allow the observational data to pick up the best possibilities, removing possible biases due to the flatness assumption.
Moreover, we are also interested in investigating what happens to the current cosmological tensions [22]} when the curvature of the Universe is allowed into the picture. In particular, we want to see what happens to the value of the Hubble constant and the \(>5\sigma \)  tension existing between the \(\Lambda \) CDM based Planck 2018 estimate [8]} and the SH0ES (Supernovae and \(H_0\)  for the Equation of State of dark energy) measurement [24]}, [25]}, known as the Hubble tension [26]}, [27]} (see also Refs. [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}, [38]}, [39]}, [40]}, [41]}, [42]}, [43]}, [44]}, [45]}, [46]}, [47]}, [48]}, [49]}, [50]}, [51]}, [52]}, [53]}, [54]}, [55]}, [56]}, [57]}, [58]}, [36]}, [60]}, [38]}, [37]}, [63]}, [41]}, [47]}, [66]}, [67]} aiming to alleviate the \(H_0\)  tension in various ways), together with the \(>3\sigma \)  tension in the \(S_8\)  parameter [68]} defined as a combination of the amplitude of the matter power spectrum \(\sigma _8\)  with the matter density at present \(\Omega _{m0}\)  (\(S_8 = \sigma _8\sqrt{\Omega _{m0}/0.3}\) ) between the CMB data (within the \(\Lambda \) CDM assumption) and the weak lensing experiments [69]}, [70]}, [71]}, [72]} (see Refs. [73]}, [74]}, [75]}, [76]}, [77]}, [47]}, [79]}, [80]}, [33]}, [82]}, [83]}, [56]} offering a possible route to alleviate the \(S_8\)  tension in various alternatives to \(\Lambda \) CDM).
